Aops Expressions With Fractions Art of Problem Solving Expressions With Fractions

This article explains how to use LaTeX in the AoPSWiki, the AoPS Community, and the AoPS Classroom. See Packages to know which packages are prebuilt into the AoPS site.

i Getting Started with LaTeX
- 1.1 The Very Basics
  - i.1.1 In-line Math Mode
  - 1.one.two Display Math Mode
  - 1.1.3 In-line vs. Brandish
- 1.ii Basic Expressions
  - 1.ii.one Multiplication
  - 1.2.2 Fractions
  - one.two.3 Roots
  - 1.2.4 Superscripts & Subscripts
  - 1.2.v Groups
- 1.3 Beyond the Basic Expressions
  - ane.3.1 Grouping Basic Expressions
  - 1.3.ii Lists
  - 1.iii.3 Sums
  - 1.3.4 Products
- 1.iv Equalities and Inequalities
  - 1.4.1 Inequalities
  - ane.4.2 Aligning Equations
  - 1.4.3 Numbering Equations
  - 1.4.4 Comments in Equations
  - 1.4.5 Definition by Cases

Getting Started with LaTeX

The Very Basics

LaTeX uses a special "math fashion" to display mathematics. At that place are ii types of this "math way":

In-line Math Mode

In in-line math mode, we apply $ signs to enclose the math we want to display, and it displays in-line with our text. For example, typing $\sqrt{x} = 5$ gives us $\sqrt{x} = 5.$

Display Math Mode

In display math mode, we enclose our code in double dollar signs, and it displays the math centered and on its own line. For instance, $$\sqrt{ten} = 5$$ gives the states $\sqrt{x} = 5.$

In-line vs. Display

Besides displaying in-line vs. displaying centered and on a new line, the two modes render differently in other ways. Note that $\sum_{one thousand=1}^n k^2$ gives us $\textstyle\sum_{k=1}^n k^2,$ whereas $$\sum_{k=1}^n k^2$$ gives united states $\sum_{k=1}^n k^2.$

Bones Expressions

Multiplication

Sometimes, when nosotros're multiplying, we don't need a multiplication symbol. For instance, we can write $xy$ instead of $x\cdot y$ without ambiguity. All the same, when you're multiplying numbers, for instance, a multiplication symbol comes in handy. The standard symbol is given by $\cdot$ . For example, $12\cdot\frac{1}{2}$ gives us $\textstyle 12\cdot \frac 12.$

Fractions

We can make fractions via $\frac{...}{...}$ . For case, $\frac{x+y}{2}$ volition give us $\textstyle\frac{x+y}{2}.$

Roots

Square roots in $\LaTeX$ are pretty simple; nosotros simply type $\sqrt{...}$ . For example, $\sqrt{two}$ gives united states of america $\sqrt 2.$ Cube roots, fourth roots, and so on are only slightly more than hard; we type $\sqrt[north]{...}$ . For instance, $\sqrt[4]{x-y}$ gives $\sqrt[4]{x-y}.$

Superscripts & Subscripts

To get superscripts (or exponents), we use the caret symbol ^. Typing $x^two+y^2$ gives $x^2+y^2.$ Subscripts are obtained via an underscore (belongings shift and the minus sign on most keyboards). For instance, $a_k$ yields $a_k.$

Groups

About operations in $\LaTeX$ (such equally superscripts and subscripts) can only encounter the "group" of characters immediately following it. Nosotros use curly braces {...} to indicate groups longer than one character. For instance, if we wrote $x^2015$ , we'd expect to go $x^{2015},$ but we instead become $x^2015.$ This is because each graphic symbol in the string 2015 is in its ain group until we tell $\LaTeX$ that 2015 should be one whole grouping. To convey this information to $\LaTeX$ , we write $x^{2015}$ and we get $x^{2015}.$

Across the Basic Expressions

Grouping Basic Expressions

Our ordinary parentheses (...) and brackets [...] work to group expressions in $\LaTeX$ . For example, $(10+y)[z+w]$ gives u.s.a. $(x+y)[z+w].$ We can also group expressions using curly braces, but we can't just blazon {...}. Rather, we must type \{...\}. This is because $\LaTeX$ uses plain curly braces for other things, such equally fractions and superscripts and subscripts.

When we put (vertically) large expressions within of parentheses (or brackets, or curly braces, etc.), the parentheses don't resize to fit the expression and instead remain relatively pocket-sized. For instance, $$f(x) = \pi(\frac{\sqrt{ten}}{x-1})$$ comes out as $f(x) = \pi(\frac{\sqrt{x}}{x-1}).$ To automatically adjust the size of parentheses to fit the expression inside of them, we type \left(...\right). If we exercise this for our $f$ equation above, we get $f(x) = \pi\left(\frac{\sqrt{x}}{x-1}\right).$ We tin can use \left and \right for all sorts of things... parentheses (as nosotros saw), brackets $\left[...\right]$ , braces $\left\{...\right\}, absolute values $\left|...\right|$ , and much more (norms, floor and ceiling functions, inner products, etc.).

Lists

To make a list, such as a sequence, we use \dots. For case, $a_0,a_1,\dots,a_n$ volition requite us $a_0,a_1,\dots,a_n.$

Sums

In that location are 2 bones ways to write out sums. First, we can use + and \cdots. An example of this style would be $a_1+a_2+\cdots+a_n$ This will requite usa $a_1+a_2+\cdots+a_n.$ Second, nosotros could use summation notation, or \sum. Such an example is $\sum_{i=0}^due north a_i$ , giving $\textstyle \sum_{i=0}^n a_i.$ Note the use of superscripts and subscripts to obtain the summation alphabetize.

Products

Once more, in that location are two basic ways to brandish products. First, we can utilise \cdot and \cdots. An case is $n! = northward\cdot(n-ane)\cdots two\cdot i$ , which of course gives $n! = n\cdot(n-1)\cdots 2 \cdot 1.$ The culling is to employ product note with \prod. For instance, $northward! = \prod_{thousand=ane}^due north k$ , giving $\textstyle n! = \prod_{k=1}^n k.$

Equalities and Inequalities

Inequalities

the commands >, <, \geq, \leq, and \neq give us $>,$ $<,$ $\geq,$ $\leq,$ and $\neq,$ respectively.

Adjustment Equations

To align multiple equations, we apply the align* surroundings. For instance, nosotros might blazon a system of equations as follows:

\begin{align*} ax + by &= 1 \\ cx + dy &= 2 \\ ex + fy &= 3. \cease{align*}

(You do non need dollar signs.) The & symbol tells $\LaTeX$ where to align to and the \\ symbols break to the next line. This code will output $\begin{align*} ax + by &= 1 \\ cx + dy &= 2 \\ ex + fy &= 3. \end{align*}$ An instance of a string of equations is:

\begin{align*} ((2x+3)^iii)' &= iii(2x+3)^2 \cdot (2x+3)' \\ &= 3(2x+three)^2 \cdot 2 \\ &= half-dozen(2x+3)^two. \end{marshal*}

Again, the & symbol tells $\LaTeX$ where to align to, and the \\ symbols pause to the next line. This lawmaking outputs $\begin{align*} ((2x+3)^3)' &= 3(2x+3)^2 \cdot (2x+3)' \\ &= 3(2x+3)^2 \cdot 2 \\ &= 6(2x+3)^2. \end{align*}$

Numbering Equations

To number equations, nosotros utilise the align environs. This is the aforementioned environment equally the align* environment, just nosotros go out the * off. The * suppresses numbering. To number one equation, the code

\begin{align} ax + past = c \finish{marshal}

will produce $\begin{align} ax + by = c. \end{align}$ Nosotros don't accept to employ & or \\ since there is nix to align and no lines to interruption. To number several equations, such as a system, the code

\begin{align} ax + past &= c \\ dx + ey &= f \\ gx + hy &= i \finish{align}

will produce $\begin{align} ax + by &= c \\ dx + ey &= f \\ gx + hy &= i. \end{align}$ In general, align will automobile-number your equations from kickoff to last.

Again, we use the align* environs. The lawmaking

\begin{marshal*} ax + past &= c & \text{considering apathetic} \\ dx + ey &= f & \text{past such-and-such} \end{marshal*}

volition produce $\begin{align*} ax + by &= c & \text{because blah} \\ dx + ey &= f & \text{by such-and-such}. \end{align*}$ (You lot can use marshal to become numbering and comments!)

Definition by Cases

To define, say, a office by cases, nosotros use the cases surroundings. The lawmaking

$$  \delta(i,j) = \begin{cases} 0 & \text{if } i \neq j \\ 1 &\text{if } i = j \end{cases}  $$

gives united states $\delta(i,j) = \begin{cases} 0 & \text{if } i \neq j \\ 1 &\text{if } i = j. \end{cases}$ As usual, the & is for aligning and the \\ is for line-breaking.

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Source: https://artofproblemsolving.com/wiki/index.php/LaTeX:LaTeX_on_AoPS